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人気のある三角関数 >

8cos^2(x)+6cos(x)+1=0

解

8 cos^{2} (x) + 6 cos (x) + 1 = 0

解

x = 1.82347 \dots + 2 πn, x = - 1.82347 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

+1

度

x = 104.47751 \dots^{\circ} + 36 0^{\circ} n, x = - 104.47751 \dots^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

解答ステップ

8 cos^{2} (x) + 6 cos (x) + 1 = 0

置換で解く

8 cos^{2} (x) + 6 cos (x) + 1 = 0

仮定:

cos (x) = u

8 u^{2} + 6 u + 1 = 0

8 u^{2} + 6 u + 1 = 0 : u = - \frac{1}{4}, u = - \frac{1}{2}

8 u^{2} + 6 u + 1 = 0

解くとthe二次式

8 u^{2} + 6 u + 1 = 0

二次Equationの公式:

次の場合:

a = 8, b = 6, c = 1

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 8 \cdot 1}{2 \cdot 8}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 8 \cdot 1}{2 \cdot 8}

6^{2} - 4 \cdot 8 \cdot 1 = 2

6^{2} - 4 \cdot 8 \cdot 1

数を乗じる:

4 \cdot 8 \cdot 1 = 32

= 6^{2} - 32

6^{2} = 36

= 36 - 32

数を引く:

36 - 32 = 4

= 4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

u_{1, 2} = \frac{- 6 \pm 2}{2 \cdot 8}

解を分離する

u_{1} = \frac{- 6 + 2}{2 \cdot 8}, u_{2} = \frac{- 6 - 2}{2 \cdot 8}

u = \frac{- 6 + 2}{2 \cdot 8} : - \frac{1}{4}

\frac{- 6 + 2}{2 \cdot 8}

数を足す/引く:

- 6 + 2 = - 4

= \frac{- 4}{2 \cdot 8}

数を乗じる:

2 \cdot 8 = 16

= \frac{- 4}{16}

分数の規則を適用する:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4}{16}

共通因数を約分する:

4

= - \frac{1}{4}

u = \frac{- 6 - 2}{2 \cdot 8} : - \frac{1}{2}

\frac{- 6 - 2}{2 \cdot 8}

数を引く:

- 6 - 2 = - 8

= \frac{- 8}{2 \cdot 8}

数を乗じる:

2 \cdot 8 = 16

= \frac{- 8}{16}

分数の規則を適用する:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{8}{16}

共通因数を約分する:

8

= - \frac{1}{2}

二次equationの解:

u = - \frac{1}{4}, u = - \frac{1}{2}

代用を戻す

u = cos (x)

cos (x) = - \frac{1}{4}, cos (x) = - \frac{1}{2}

cos (x) = - \frac{1}{4}, cos (x) = - \frac{1}{2}

cos (x) = - \frac{1}{4} : x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

cos (x) = - \frac{1}{4}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{1}{4}

以下の一般解

cos (x) = - \frac{1}{4}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (x) = - \frac{1}{2}

以下の一般解

cos (x) = - \frac{1}{2}

cos (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

すべての解を組み合わせる

x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

10進法形式で解を証明する

x = 1.82347 \dots + 2 πn, x = - 1.82347 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

グラフ

人気の例

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